The definition of a double-scaling limit represents an important goal in the development of tensor models.We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the iid tensor models.We show that the radius of convergence of the nlo series coincides with that of the leading order melonic sector.Meanwhile, the value of the susceptibility exponent, γnlo = 3/2, signals a departure from the leading order behaviour.Both pieces of information provide clues for a non-trivial double-scaling limit, for which we put forward some precise conjecture.
In this paper we study the double scaling limit of the multi-orientable tensor model. We prove that, contrary to the case of matrix models but similarly to the case of invariant tensor models, the double scaling series are convergent. We resum the double scaling series of the two-point function and of the leading singular part of the four-point function. We discuss the behavior of the leading singular part of arbitrary correlation functions. We show that the contribution of the four-point function and of all the higher point functions are enhanced in the double scaling limit. We finally show that all the correlation functions exhibit a singularity at the same critical value of the double scaling parameter which, combined with the convergence of the double scaling series, suggest the existence of a triple scaling limit.
Abstract This chapter presents the double scaling limit of quartic tensor models. In the first part of this chapter it is shown show that the melonic family in the quartic melonic tensor model can be analytically resumed and that this resummation is encoded in a translation of the intermediate matrix fields to a nontrivial vacuum. The most striking consequence of this translation is that one can now show that the continuum limit of tensor models corresponds to a genuine phase transition (in the field theory sense) associated to a symmetry breaking. In the second part of this chapter the intermediate field representation is used to derive the double scaling limit of the quartic melonic model.
Abstract In Giombi et al (2018 Phys. Rev. D 98 105005), a prismatic tensor model was introduced. We study here the diagrammatics and the double scaling limit of this model, using the intermediate field method. We explicitly exhibit the next-to-leading order Feynman graphs in the 1 / N expansion. Using appropriate combinatorial tools, we further study the general term of the 1 / N expansion and we compute the two-point function in the double scaling limit. We find that the double scaling limit mechanism implemented here is similar to the one implemented for various quartic tensor models.
This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with
O(N)3 symmetry and long-range propagator. The existence of a non-perturbative fixed point in any
d at large
N is established. We found four lines of fixed points parametrized by the so-called tetrahedral coupling. One of them is infrared attractive, strongly interacting and gives rise to a new kind of CFT, called melonic CFTs which are then studied in more details. We first compute dimensions of bilinears and OPE coefficients at the fixed point which are consistent with a unitary CFT at large
N. We then compute
1/N corrections. At next-to-leading order, the line of fixed points collapses to one fixed point. However, the corrections are complex and unitarity is broken at NLO. Finally, we show that this model respects the
F-theorem. The next part of the thesis investigates sextic tensor field theories in rank
3 and
5. In rank
3, we found two IR stable real fixed points in short range and a line of IR stable real fixed points in long range. Surprisingly, the only fixed point in rank
5 is the Gaussian one. For the rank
3 model, in the short-range case, we still find two IR stable fixed points at NLO. However, in the long-range case, the corrections to the fixed points are non-perturbative and hence unreliable: we found no precursor of the large
N fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large
N limit. We prove that models with tensors in an irreducible representation of
O(N) or
Sp(N) in rank
5 indeed admit a large
N limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.
Abstract In the first section we give a briefly presentation of the U(N)D-invariant tensor models (N being again the size of the tensor, and D being the dimension). The next section is then dedicated to the analysis of the Dyson–Schwinger equations (DSE) in the large N limit. These results are essential to implement the double scaling limit mechanism of the DSEs, which is done in the third section. The main result of this chapter is the doubly-scaled 2-point function for a model with generic melonic interactions. However, several assumptions on the large N scaling of cumulants are made along the way. They are proved using various combinatorial methods.